| L(s) = 1 | − 0.832i·5-s + 2.52·7-s + 10.9i·11-s − 8.72·13-s + 20.8i·17-s − 1.50·19-s + 1.15i·23-s + 24.3·25-s − 18.1i·29-s − 51.3·31-s − 2.09i·35-s − 7.93·37-s − 25.2i·41-s + 38.6·43-s + 68.8i·47-s + ⋯ |
| L(s) = 1 | − 0.166i·5-s + 0.360·7-s + 0.994i·11-s − 0.671·13-s + 1.22i·17-s − 0.0790·19-s + 0.0503i·23-s + 0.972·25-s − 0.626i·29-s − 1.65·31-s − 0.0599i·35-s − 0.214·37-s − 0.616i·41-s + 0.899·43-s + 1.46i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.2797221789\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2797221789\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 0.832iT - 25T^{2} \) |
| 7 | \( 1 - 2.52T + 49T^{2} \) |
| 11 | \( 1 - 10.9iT - 121T^{2} \) |
| 13 | \( 1 + 8.72T + 169T^{2} \) |
| 17 | \( 1 - 20.8iT - 289T^{2} \) |
| 19 | \( 1 + 1.50T + 361T^{2} \) |
| 23 | \( 1 - 1.15iT - 529T^{2} \) |
| 29 | \( 1 + 18.1iT - 841T^{2} \) |
| 31 | \( 1 + 51.3T + 961T^{2} \) |
| 37 | \( 1 + 7.93T + 1.36e3T^{2} \) |
| 41 | \( 1 + 25.2iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 38.6T + 1.84e3T^{2} \) |
| 47 | \( 1 - 68.8iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 46.5iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 102. iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 88.3T + 3.72e3T^{2} \) |
| 67 | \( 1 + 22.6T + 4.48e3T^{2} \) |
| 71 | \( 1 + 104. iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 75.2T + 5.32e3T^{2} \) |
| 79 | \( 1 - 103.T + 6.24e3T^{2} \) |
| 83 | \( 1 - 62.0iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 1.95iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 118.T + 9.40e3T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.211265252017805586840193072430, −8.214404685847028919743143788437, −7.62515714680995720704014669152, −6.86173768810247844913682357071, −5.99980059136094838986967626430, −5.06342126662355035175387943881, −4.44212113286540848003998368430, −3.50625426613519406456793072506, −2.27676870552868203902573217232, −1.49780312972647529081046037878,
0.06434818735268672757909301150, 1.28995232984659540889772957903, 2.58611050954368057664360660493, 3.28571318575708496882507668006, 4.41493322603006516038952621437, 5.22413325288899517902175070887, 5.88796779794047603359779058836, 7.01702668276124018503603047220, 7.39127383978954496838946267359, 8.424935950817085136211634069890
